Classifying toposes for first-order theories
type: Item
instance of: scholarly article
title: Classifying toposes for first-order theories
zbMATH Open document ID: 0893.03027
DOI: 10.1016/S0168-0072(97)00042-0
author: Carsten Butz
author: Peter Johnstone
published in: Annals of Pure and Applied Logic
publication date: 1998-07-01T00:00:00Z
review text: We say that a (Grothendieck) topos \(\mathcal E\) is a classifying topos for an infinitary first-order theory \(\mathbf T\) if, for any topos \(\mathcal F\), there is an equivalence between the category of \(\mathbf T\)-models in \(\mathcal F\) and the category of open geometric morphisms \({\mathcal F}\to{\mathcal E}\), which is `natural in \(\mathcal F\)' in an appropriate sense. In this paper we characterize the first-order theories which have classifying toposes, as those for which the Lindenbaum algebra of provable-equivalence classes of \({\mathcal L}_{\infty\omega}\) formulae in any context is small; we also show that every Grothendieck topos occurs as a classifying topos in this sense, and that every geometric theory has a canonical geometrically-conservative extension to a first-order theory with this smallness property. We give an explicit presentation of this extension for a couple of simple theories; but we also show, using a result of D. de Jongh, that `most' familiar first-order theories do not satisfy the smallness condition.
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 03C75
Mathematics Subject Classification ID: 18B25
zbMATH DE Number: 1133083
zbMATH Keywords: classifying topos for an infinitary first-order theory
zbMATH Keywords: Grothendieck topos
zbMATH Keywords: geometric theory
zbMATH Keywords: smallness condition for Lindenbaum algebras
reviewed by: Peter Johnstone
MaRDI profile type: Publication
cites work: Boolean classifying topoi
cites work: La logique des topos
cites work: Change of base for toposes with generators
cites work: Q3050433
cites work: Q3995720
cites work: Infinite Boolean polynomials I
cites work: On the non-existence of free complete Boolean algebras
cites work: Q4145861
cites work: Open maps of toposes
cites work: Q3965241
cites work: Q3936721
cites work: An extension of the Galois theory of Grothendieck
cites work: Infinitary propositional intuitionistic logic
cites work: Universal projective geometry via topos theory
cites work: Sheaves in geometry and logic: a first introduction to topos theory
cites work: Completeness results for intuitionistic and modal logic in a categorical setting
cites work: Minimal models of Heyting arithmetic
cites work: Infinitary intuitionistic logic from a classical point of view
cites work: Constructive Sheaf Semantics
cites work: New proof of a theorem of Gaifman and Hales
full work available at URL: s0168-0072(97)00042-0
OpenAlex ID: W2088933768
description: scientific article; zbMATH DE number 1133083